One - which I won't elaborate on, unless and until I find out about archaeological finds - is the so-called Elamite dagger, shown in the Persepolis reliefs and on the Egyptian statue of Darius the Great. Unfortunately I've only ever seen it in sculpture, invariably sheathed, so there's very little to tell about what it looked like.
The most well-known type associated with the Persians was called in Greek the akinakes (sometimes akenakes, unattested Persian *akinaka, Latin acinaces). Because the word means a Persian sword, in early modern times it was taken to refer to a type of saber or scimitar, and by extension other curved swords like the katana.
In truth the Classical akinakes was a short, straight, double-edged weapon, ranging in size from a small dagger to almost as big as a gladius. The type is closely associated with the culture of the Great Steppe; variations were used from the Ukraine to western China. While Steppe examples could still be bronze in this period, the Persian akinakes was made of iron. The guard was typically shaped like an upside-down heart, and the columnar grip and rounded bar pommel formed a T. Ornate golden examples exist which differ in many aspects.
The easiest way to create an akinakes is to hilt or rehilt a dagger blade of appropriate shape. While metal hilts are attested, XMFM will approve of the use of organic hilts. Windlass Steelcrafts and its distributors sell a big and cheap Arkansas toothpick blade which, while perhaps too tapered, is otherwise perfect for the part. If you can find other examples, use them; just avoid undocumented fullering patterns and ricassos.
If you have the material skills, peen the tang over a washer. Threaded tangs are okay, but to avoid the glaring anachronism, the securing nut should be either a tang nut sunken into the pommel, or low in profile and round, or a square or hex nut ground down as much as possible. Unground ones are okay on sparring blades, which will never be entirely realistic anway.
Sparring blades are generally light and flexible and must have either very obtuse rounded points or rolled ones. Some options may include Darkwood's Wide Flex, Alchem's Safeflex and Hanwei's Practical Main Gauche blade (available from many distributors).
The akinakes was worn at the right hip in an elaborate scabbard. The chape, generally a rounded triangle, could be made of bronze, chased gold or carved ivory, and probably lots of other materials. Just above the chape, a cord was tied around the scabbard, passed around the thigh and then through a slipknot next to the chape. The throat had a large tab, which at its own upper corner was tied through a hole in the weapon belt. The throat should completely cover the sword's guard, but I haven't worked out a way of doing this yet. This is a very elegant and handy suspension system for a very small weapon.
The last type(s) of sword was the single-edged, belly-bladed sword, in various forms called a kopis or machaira. It was Greek in origin but in Greek art often shown wielded by Persian or Scythian warriors. It was a somewhat longer sword than the akinakes. Its shape gave it great cutting power (indeed kopis is Greek for "cutter"). It often had a single flared shoulder, on the sharp or forward side of the blade; and normally a full-profile or slab tang and a pommel which was simply a swelling of the tang on the forward side. The pommel might be bulbous (the grip might then be carved into a curl, like a fiddlehead), or hooked and resemble a bird's head in profile. Some kopides had a full S-shaped crossguard.
No one, to my knowledge, makes an off-the-shelf kopis that's historically-accurate yet inexpensive. Windlass makes a small one of dubious historicity (it would require rehilting at the very least). Their Cobra Steel kopis is a modern interpretation and the slab tang is the wrong shape. Windlass, Del Tin, Deepeeka and possibly others have historical falcatas in their lineup of varying accuracy, but these are Iberian-style swords which only bear the broadest resemblance to Eastern Mediterranean ones. If you have enough money, getting a custom blade is the best option.
Next up: How to make a scabbard.